Lecture on Mutually Exclusive Events

Definition

• Mutually Exclusive Events: Events that cannot occur simultaneously.
  • If two events are mutually exclusive, they do not share any outcomes.
  • Probability of both events occurring together is zero.

Examples

• Rolling a six-sided die:
  • Sample space: {1, 2, 3, 4, 5, 6}
  • Event A: {1, 2, 3}
  • Event B: {5, 6}
  • Event C: {3, 4, 5}

Analysis of Events

1. Events A and B
   • No shared outcomes.
   • Probability of A and B: 0
   • Mutually Exclusive
2. Events A and C
   • Shared outcome: 3
   • Probability of A and C: 1/6
   • Not Mutually Exclusive
3. Events B and C
   • Shared outcome: 5
   • Probability of B and C: 1/6
   • Not Mutually Exclusive

Venn Diagrams

• Illustration:
  • Mutually exclusive events do not share any part in a Venn diagram.
  • Non-mutually exclusive events share an intersection.

Probability Calculations

Events that are Not Mutually Exclusive

• Formula:
  • ( P(A \cup B) = P(A) + P(B) - P(A \cap B) )
  • Used for calculating the probability of either event A or B occurring.
  • Ensures not double-counting shared outcomes.

Example Calculation

• Event A: {1, 2, 3, 4}
• Event B: {3, 4, 5}
• Probability Calculation:
  • ( P(A) = \frac{4}{6} )
  • ( P(B) = \frac{3}{6} )
  • ( P(A \cap B) = \frac{2}{6} )
  • ( P(A \cup B) = \frac{5}{6} )

Events that are Mutually Exclusive

• Simplified Formula:
  • ( P(B \cup C) = P(B) + P(C) )
• Example with Events B and C (mutually exclusive):
  • ( P(B) = \frac{3}{6} )
  • ( P(C) = \frac{1}{6} )
  • ( P(B \cap C) = 0 )
  • ( P(B \cup C) = \frac{2}{3} )

Key Takeaways

• Mutually Exclusive Events: No shared outcomes, probability of intersection is zero.
• Probability Calculation: Adjust formulas based on whether events are mutually exclusive.
• Understanding through Venn Diagrams: Visual representation can help in identifying shared outcomes.